how to prove the Abel's test for uniform convergence?

The Abel's test for uniform convergence states: Assume converges uniformly on E. A uniformly bounded sequence of functions satisfies for all n and x in E. Then converges uniformly.
My tentative proof is as follows:
Let . By the partial summation formula, we have . So if n>m we can get . If the uniform bound for is M, the absolute value of the above difference satisfies . From the fact that converges uniformly, for any given , there is an N such that for all m,n>N we have . So the right side of the above inequality . But the variable (n-m) makes the Cauchy condition inapplicable. Can you tell me how to proceed, or is there any other better proof? Thanks!

the absolute difference should go as follows:
For . Now the Cauchy condition can be applied.
Another proof can be found in K. Knopp's "Theory and application of infinite series", P346-347. This proof is more complicated than the above one.